In this review, we did not detail a last equivalence of the static formulation when the cost is a metric \(d_\) , which corresponds to \(W_1\)  OT in the balanced case. In that case one can reformulate it as an integral probability metric [1]} over the set of Lipschitz functions. It is also equivalent to a specific dynamic formulation called Beckmann. We refer to [2]} for more details on the balanced OT setting. For the unbalanced extension of the Lipschitz formulation, see [3]}, [4]}, [5]}, and we refer to [5]} for the generalized Beckmann problem.
